Unitary, continuum, stationary perturbation theory for the radial Schr\"odinger equation

TL;DR

This paper develops a stationary perturbation theory for the radial Schrödinger equation using unitary transformations, providing a systematic way to compute phase shifts with verified accuracy against exact solutions.

Contribution

It introduces a novel unitary, continuum, stationary perturbation framework for the radial Schrödinger equation, extending the concept of unitary transformations to nonrelativistic scattering problems.

Findings

First- and second-order phase shifts match exact solutions.

Method improves upon standard perturbation theory.

Complete agreement with known solutions for test cases.

Abstract

The commutators of the Poincar\'e group generators will be unchanged in form if a unitary transformation relates the free generators to the generators of an interacting relativistic theory. We test the concept of unitary transformations of generators in the nonrelativistic case, requiring that the free and interacting Hamiltonians be related by a unitary transformation. Other authors have applied this concept to time-dependent perturbation theory to give unitarity of the time evolution operator to each order in perturbation theory, with results that show improvement over the standard perturbation theory. In our case, a stationary perturbation theory can be constructed to find approximate solutions of the radial Schr\"odinger equation for scattering from a spherically symmetric potential. General formulae are obtained for the phase shifts at first and second order in the coupling constant. We test the method on a simple system with a known exact solution and find complete agreement between our first- and second-order contributions to the s-wave phase shifts and the corresponding expansion to second order of the exact solution.